There’s an old labor joke, there are three labor laws in the USA. Federal, state and California.
In general, unless you are covered by a CBA or legally binding employment agreement, your employer can make you do anything that isn’t illegal.
An employer can certainly make you work overtime. If you are non-exempt they must pay you overtime. Exempt employees may or may not (usually not) get overtime or comp time. (Note:Some federal employees are exempt from overtime and may get comp time instead. Not so with private workers).
Most states and the federal government have no provision for “on-call” pay. If your state does not, you can be required to be “on-call” 24/7.
Some states have provisions limiting the number of days you can work in a row. For instance, Illinois requires you have at least one day, in a Sun-Sat work week off. But remember your employer determines your work week.
So in IL, you could be scheduled the first week, Tue-Sat (5 days) then Sunday starts a new work week, so then you could be scheduled Sun-Thu (5 days) and wind up working ten days in a row.
In regards to using the FMLA, the taking off of Fridays and Mondays has been used so often they amended the act to give employers more power to force the employee to prove he really needs the Monday or Friday off. This was due to a lot of abuse of what is termed intermittent FMLA use. Which IS perfectly legal, if really needed.
The original FMLA examp is paid for by the employee. If the employer requires a second exam to verify the employees need, then the employer pays for it.
If that second opinion is contrary to the first, the employer can request a third exam, again paid for by the employer which is final.
Remember the FMLA (and even the ADA) requires your employer make a reasonable accommodation. It does not have to be the accommodation you want, or even the one your doctor wants, just that it accommodates you within reason.
To be eligible for FMLA, you have to work for your employer for the last 12 months, worked at least 1,250 hours, at a location where the employer has 50+ employees, within 75 miles. (this would cover multiple offices).
The fact that non of you signed up has little meaning, as employers are free to change job descriptions and requirements at will, depending on business needs, provided they don’t do so for discriminatory reasons, of a protected class.
I will add California has a lot more restrictive laws and protections for workers, and you would have to see the DOL website or offices if the OP lives in that state.
As an exempt employee I found myself in a similar position where I had a lot of weekends I had to work as we couldn’t find an employer to cover the position, so I sympathise.
